Green's symmetries in finite digraphs (Q527577)

Summary: The semigroup \texttt{D}\(_V\) of digraphs on a set \(V\) of \(n\) labeled vertices is defined. It is shown that \texttt{D}\(_V\) is faithfully represented by the semigroup \(B_n\) of \(n\times n\) Boolean matrices and that the Green's \texttt{L, R, H}, and \texttt{D} equivalence classifications of digraphs in \texttt{D}\(_V\) follow directly from the Green's classifications already established for \(B_n\). The new results found from this are: (i) \texttt{L, R}, and \texttt{H} equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green's \textit{iso neighborhood symmetries}; and (ii) \texttt{D} equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same \texttt{D} equivalence class, i.e., these digraphs are characterized by Green's \textit{isolattice symmetries}. As a simple illustrative example, the Green's classification of all digraphs on two vertices is presented and the associated Green's symmetries are identified.